growth rate

max numbers

angle spread

growth exponent

trunk height

primes

highly composites

branches

distance arcs

numbers

order rays

Each number branches from its largest divisor

The integers have a beautiful fractal structure that can be seen by sorting them in terms of divisibility. The Number Tree is constructed using a simple rule: every integer (except 0 and 1) branches off from its largest divisor. For example, consider the number 20. It has five divisors (not counting itself): 1, 2, 4, 5, and 10. 10 is the largest divisor, and so the number 20 branches from the number 10. Now consider the number 21. It has 1, 3 and 7 as its divisors, and so it branches from 7. The number 22 branches from 11. And the number 23 (a lonely prime) branches from 1.

Notice how a fully-grown tree is bushier on the left, and a spiral of prime numbers extends out at lower-right. This is due to the ordering of the numbers, which is expressed in terms of angles.

The position of a given number (

Every new number is inserted at an order-position adjacent to its largest divisor (one order-unit counter-clockwise). Using this simple rule, patterns emerge as the tree grows. Branch-points represent multiplication by primes. The highly-composite numbers are concentrated at the lower-left, as well as power series (such as the unbranchable 2

created by Jeffrey Ventrella ( Ventrella.com )